Sections 6.1 and 6.2: Systems of Linear Equations
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1 What is a linear equation? Sections 6.1 and 6.2: Systems of Linear Equations We are now going to discuss solving systems of two or more linear equations with two variables. Recall that solving an equation in one variable means to find a value of the variable that makes the equation true. For a system of linear equations in two variables, we need to find the values of both variables that would make both equations true. Determine whether or not (3, -2) is a solution to this system. 2x y = 8 x + 3y = 4 To solve such a system, we need to look for a point in (x, y) form that makes both equations true. Substitution Method 1. Choose one equation and solve for one of the variables. 2. Substitute the result of step 1 into the equation that was not used in step 1. This should give you an equation in one variable, which you should then solve. 3. Take your answer from step two and plug it into either equation from the system. This should give you an equation that you can use to solve for the remaining variable. 4. Write your answer as a point. 4x + 2y = -8 3x 5y = 7 4x y = 17-8x + 2y = -34 1/5 Sections6.1and6.2.pdf (#19)
2 When using the addition method (sometimes called the elimination method), we want to eliminate one variable from the system so that we then have an easy equation to solve. If the x-coefficients or y-coefficients are opposites, then we can eliminate a variable by adding the equations. If the system doesn t contain one of the desired pairs of opposite coefficients, we should begin by multiplying one (or both) of the equations by constants, with the goal of obtaining opposites as coefficients for the x terms or the y terms. x 3y = -2-3x + 9y = 5 4x + 5y = 14 3x 2y = -1 Why is it helpful to study systems with three variables instead of two? A point in a three-dimensional coordinate system can be represented as an ordered triple: (x, y, z) What does the equation Ax + By = C represent? What does the equation Ax + By + Cz = D represent? Solving a system of three equations with three variables means finding a point where three planes intersect. See p. 792, Figure 7.9 2/5 Sections6.1and6.2.pdf (2/5)
3 To solve a system with three variables and three equations, we want to reduce it to a system of two equations with two variables. To do so, eliminate the variable from two equations using substitution or addition. x + 2y - 3z = 9 2x - y + 2z = -8 -x + 3y - 4z = 15 Matrices are two-dimensional arrays that can be used to solve a variety of problems. An M x N matrix has M rows and N columns. The matrices that we will be dealing with will have numbers for each entry. How many rows and columns are there in a 4 x 5 matrix? A system of linear equations can be represented as a matrix, with one row for each equation and one column for each variable, with an additional column for the constant of each equation. Each matrix entry will be a numerical coefficient of a variable or a numerical constant. 3/5 Sections6.1and6.2.pdf (3/5)
4 2x + 3y = 7 4x - 5y = 3 x + 2y - 3z = -4 3x + y + 2z = 11 2x + 3y - z = 5 When solving systems of equations with matrices, the goal is to use the above operations to change a matrix to its reduced row-echelon form. We will get the calculator to do the work for us. To go to the TI-84 calculator s matrix menu, press 2 nd and x Go to the matrix menu and move to the EDIT column. Select a number for the matrix you wish to edit. 2. Make sure that the matrix has the appropriate number of rows (number of equations) and columns (number of variables plus one) 3. Enter the coefficients of each equation. Remember to include a zero if a variable isn t part of an equation. The last entry in a row is the constant. 4. Press 2 nd and MODE to quit and go back to the normal screen. 5. Go to the matrix menu and move to the MATH column. Select rref (Note: do not select ref ) 6. Go to the matrix menu and select the matrix that contains your entries. Then press ENTER. The final column contains the values of your solution (with some exceptions that we will go over below). Remember to write your solutions as ordered pairs or ordered triples. 2x + 3y + 4z = 3 3x + 2y + 5z = 3 2x + 3y + 4z = 3 2x - 4y + 6z = 3 4/5 Sections6.1and6.2.pdf (4/5)
5 2x + 3y + 4z = 3 2x - 4y + 6z = -6 x + 2y - z = 4 w + 3x + y = 10 3w + y - 2z = -2 2w + 2x + 2z = 14 Practice Problems p. 275, problems 3-16 (use either method, show your work) p. 287, problems (use any method, calculator is easiest) 5/5 Sections6.1and6.2.pdf (5/5)
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